Status update:
I disconnected and took everything out of the computer (RAM, video card, drives, TV tuner, etc). I put a temporary start switch in the PSU and checked the voltages. They all checked out fine. Hooked up just the motherboard and fired it up. I now get one long continuous beep repeated over and over until I shut off the PSU.
So unless I'm mistaken, according to
these beep codes, that means I have a RAM problem of some sort. The weird thing is that if I put anything back in, like the video card or a single stick of RAM, I get no beeps. I would expect to get a single beep saying the system is fine or a different beep code saying that there's some other problem (like missing display card).
I think my next step is that I'm going to try to reset the cmos. Maybe it's just goobered up somehow. Tomorrow I'll see if I can borrow a stick of RAM and see if that fixes the problem.
EDIT: I tried the CMOS reset and had no different results. Currently all I can do is get the motherboard to beep at me in long continuous beeps as long there is no RAM, drives, PCI cards, or PCI-e video card installed.
Also, I've noticed a bit of "whining" coming out of the PSU that I never noticed before. It's faint and I don't usually have my head down this close to the machine.
Oh, and the GPU fan sounds sick, but still seems to be spinning.